Dynamics of a delayed rumor spreading model with discontinuous threshold control

In this paper, we studied a delayed rumor spreading model with discontinuous threshold control. First, we studied the existence of equilibria of the subsystem. Regarding the delay as bifurcating parameter, the local asymptotic stability and Hopf bifurcation of the positive equilibrium are discussed by analyzing the corresponding characteristic equations of linearized systems. Then, we studied the existence of the sliding mode and analyzed the existence of the tangent equilibria, boundary equilibria, regular equilibria, and the stability of the pseudo-equilibrium. Finally, we provide some numerical simulations to verify the theoretical results.


Introduction
Since the dawn of human civilization, rumors have consistently played a vital part in the social life of people, and they are a phenomenon that cannot be ignored. In essence, rumors are unsubstantiated stories or reports that circulate within a community, typically by word of mouth or social media, and are accepted as facts, despite the fact that their original source may be unknown. As information technology continues to advance, there has been a rise in interest in discussing social networks as a new medium for communication and social bonding. Currently, rumors are spreading more quickly than at any other time in history. People experience psychological panic as a result of it, and they suffer significant economic losses [1,2]. For instance, when there was a nuclear leakage disaster in Japan, thousands of Chinese people made the error of thinking that buying iodized salt would protect them from nuclear radiation. This led to social panics and a lack of availability of table salt. It is therefore extremely important to conduct a comprehensive investigation into the rule of rumor propagation, in order to reduce rumor propagation while maintaining societal stability and security.
Understanding the characteristics of rumor dissemination and how it spreads can lead to more effective measures to preventing the spread of rumor. For this reason, the mathematical model, and in particular the epidemic model, is commonly utilized in the research of rumor dissemination in social networks. The mechanism by which rumors propagate on the web is strikingly similar to that by which infectious diseases * Corresponding author.
E-mail address: crli1976@126.com (C. Li). spread. [3,4,5,6,7]. Daley and Kendall [8] first presented the DK model of rumor propagation. Moreno et al. studied [9] the stochastic version of the MK model on scale-free networks, by means of Monte Carlo simulations. Borge-Holthoefer et al. [10] introduced two mechanisms with the aim of filling the gap between theoretical and experimental results.
Undoubtedly, understanding how to effectively stop the spread of rumors is a crucial skill for preserving social harmony. Therefore, external controls on rumor spreading have been increasingly studied by academics, as have the norms of information distribution. Zhu [11] looked into how to stop rumors from spreading in online social networks. Zhao et al. [12] introduced a propagation force into the analysis of rumor propagation and discussed rumor control strategies. Zhu et al. [13] introduced a new delayed SIR (susceptible-propagatingrecovery) epidemic-like rumor transmission model, which can be used in either a homogeneous or a heterogeneous network, and a forced silence function was introduced to discuss the control of the model. Zhu and Wang [14] studied a SAIR (susceptible-indifferent-propagatingrecovery) rumor spreading model. We refer readers to refs. [15,16,17,18] as some other related works on rumor models with control strategies.
It is noted that the control function of most models is applied at = 0. We point out that setting the beginning of control at = 0 is a very unrealistic assumption. With this in mind, a threshold control policy may be a better control strategy, which has been applied in an endemic model and a predator-prey model (see [19,20,21,22]  The following is the structure of the paper: In Section 3, some preliminaries are given. In Section 4, we conduct an analysis of the characteristic equations that correspond to each variable and explore the local stability as well as Hopf bifurcation. In Section 5, we talk about whether or not the system has a sliding domain. In Section 6, we will explain the main theoretical results through the use of numerical simulations that we will do. After then, the final portion of Section 7, which is a discussion, is presented.

Rumor spreading model formulation
We consider the threshold policy in a rumor spreading model. We divide the total people into three classes: the rumor-susceptible individuals ( ), who represent those unaware of the rumor; the rumorpropagating individuals ( ), who stand for those who believe and spread the rumor; the rumor-recovery individuals ( ), representing those who know the rumor but have ceased communicating it after meeting somebody already informed.
In the model, we assume that Λ is the constant rate of immigration of the rumor-susceptible individuals, and is the contact rate of the rumor-susceptible individuals and the rumor-propagating individuals, is the removal rate of the system, is the contact rate of the rumor-propagating peoples and the rumor-recovery peoples, that is, if the rumor-propagating peoples will become the rumor-recovery individuals after contacting with a rumor-recovery individual who tell the truth of the rumor. is the time delay from the rumor-propagating individuals to the rumor-recovery individuals.
In addition, we consider the threshold policy in the model: if the proportion of the rumor-propagating peoples is below the critical level ℎ , the control is not applied, whereas, once the ratio of the rumorpropagating peoples increases and reaches a certain level ℎ , the control strategy is implemented.
All the above parameters are all assumed positive. Fig. 1 depicts the model for the rumor spreading process based on the above facts.
In light of the above, the dynamics are governed by the system of nonlinear ordinary differential equations shown below.
where ( ) is a government control function, which is a threshold control as follows where ℎ is a threshold value. This means that only when > ℎ will the control be applied.
To best our knowledge, few researchers have examined a model of delayed rumor propagation with discontinuous threshold control. We shall examine the existence of the equilibria, its stability and bifurcation in this paper.

Preliminaries
First, we introduce some helpful attributes and definitions for Filippov systems [23].

Dynamics of subsystem 1
For subsystem 1 , the equilibria satisfy Obviously, subsystem 1 has a equilibrium 1 1 = ( , 0, 0), which is always a regular equilibrium, and it implies that both rumor-propagating individuals and rumor-recovery individuals are extinct. If = 0, from the second equation of (3), we can obtain that = . Then substitute it into the first equation of (3), we have . Therefore, if 2 < Λ , then the system has a equilibrium 2 1 = ( , Λ − 2 , 0), which implies that the rumor-recovery individuals are extinct.
Proof. (i) For the equilibrium is locally asymptotically stable for any ≥ 0, which means that rumor is wiped out in the social.
By a direction calculation, we obtain Therefore, when = 0 all the roots of Eq. (5) have a negative real part. According to the Routh-Hurwitz criterion, the equilibrium 3 1 is locally asymptotically stable when = 0. □ When > 0, let = ( > 0) be a solution of Eq. (5). Separating real and imaginary parts, we have which leads to where Obviously, we have Denote Δ = 4 2 1 − 12 2 . For the roots of (8), we have the following results.
Remark 4.4. The system has a Hopf bifurcation implying that rumor spreads in a cyclically oscillating manner, which is harmful to the stability of the social realm.

Equilibria of system (2)
In Definition 3.1-3.4, it is shown that the equilibria of the Filippov system (2) can be classified into several categories. We will abbreviate these equilibria as , , , , and for clarity. Tangent equilibrium: By Definition 3.3, the tangent equilibrium satisfies equation    . Otherwise, if = ℎ , then the system has another boundary equilibrium ) .
When = 1, from the first equation of Eq. (21) we have Substitute (22) into the second of (21), we obtain From the third equation of Eq. (21), we get Therefore, if the following conditions hold, then system (2) has a boundary equilibrium

Regular equilibrium:
To evaluate the regular and virtual equilibria of Filippov system (2), it is important to examine all of the subsystem is a regular equilibrium, while 2 3 is a virtual equilibrium.
All the cases listed above can be viewed in Fig. 2, where we define two curves can switch between the regular and the virtual equilibrium. Furthermore, it shows that the rumor spreading equilibrium constantly decreases as the parameter increases.

Existence of a sliding domain
It is well known that if two subsystems of the system (2) are oriented toward each other in Σ , a sliding domain may occur. Next, the sliding mode dynamics of the Filippov system (2) will be derived using the equivalent control method [27].
The following is a sufficient requirement for the existence of a sliding mode on a discontinuous surface From system (1), we obtain When = 0, − ℎ < 0, there must be − − > 0 in order that ( ) < 0. As a result, we have − > . When = 1, there must be − − − < 0. So, − < + . So, the sliding segment of Filippov system (2) can be written as

Sliding mode dynamics
Using the Utkin equivalent control method [27], we can derive the dynamics of the Filippov system (2) on the boundary Σ .
More specifically, ( ) = 0 and Therefore, we obtain Putting into the second equation of system (1) gives us System (23), which has a unique pseudo-equilibrium ( , ℎ , ), can be used to figure out the dynamics of Σ , where Theorem 5.1. If the pseudo-equilibrium exists, then it is locally asymptotically stable. Proof. Linearizing system (23) at the pseudo-equilibrium , we obtain the characteristic equation as follows ( + ℎ + )( + − ) = 0.
Obviously, = − ℎ − < 0. Obviously, all roots of + − = 0 have negative real parts. Therefore, the pseudo-equilibrium is locally asymptotically stable with all ≥ 0. □ Remark 5.2. The existence of sliding mode is very important for discontinuous control. We can make the system converge to any point in sliding domain Σ by selecting some value of ℎ . From the realistic perspective, it implies that we can control the rumor spread in a small range by selecting the value of ℎ .

Numerical simulations
In this part, we demonstrate our theoretical results using numerical simulations of a few situations.
6.1. The stability of 1 1 and 2 1 We choose the parameters as follows. It is easily obtained that Λ < 2 , according to Theorem 4.1, the equilibrium 1 1 = ( Λ , 0, 0) is locally asymptotically stable (Fig. 3), meaning that rumor is wiped out in the social realm.
From the above, we can obtain that has a great impact on the dynamics of system (1), and it has a threshold value . If < , then rumor cannot spread. While, if > , then rumors were spread. This indicates that when the link between rumor-susceptible and rumorspreading persons grows, it becomes easier for rumors to spread. In this context, the spread of rumors poses a significant threat to social stability.
From Theorem 5.1, we know that it is locally asymptotically stable if it exists. Furthermore, we let ℎ = 0.4, the equilibrium 3 1 is a virtual equilibrium, while 2 2 is a regular equilibrium, which is stable, see Fig. 5(c). From a realistic perspective, we choose some ℎ such that we can keep the spread of rumors in a small range through the discontinuous control.

The effect of delay
In the following, we discuss the effect of . We let the parameters be same as the above. By a direction calculation, we obtain that Eq. When = 1 , a pair of eigenvalues crosses the imaginary axis from left to right. Fig. 6 shows the delay time histories from differ- , the equilibrium of the system (2) is asymptotically stable, but it becomes unstable when ∈ ( 1 1 , 1 2 ) +∞). In other words, time delay causes the system (2) to display the phenomena of many switching events, where the state of the system changes from stable to unstable and back again. The system (2) is ultimately unstable at the equilibrium position 3 1 . Fig. 6(a), (c), (e) show that the rumor-propagating individuals converge to a positive number, implying that rumors spread in the social realm in a stable state. Fig. 6(b), (d) show that the rumor-propagating individuals maintain a fluctuating state, which means that the rumor continues to break out in stages. Fig. 6(f) shows that the rumor in the system (1) will continue to erupt periodically.
From Fig. 6 and Fig. 7, it is easily obtained that we can control the rumor in a certain range by choosing the value of ℎ . We show these in Fig. 8-Fig. 10, where ℎ = 0.1 (Fig. 8(a), Fig. 9(a), Fig. 10(a)), ℎ = 0.2 ( Fig. 8(b), Fig. 9(b), Fig. 10(b)) and ℎ = 0.4 (Fig. 8(c), Fig. 9(c), Fig. 10(c)). They show that with the increase in ℎ , the maximum value of the rumor-propagating individuals becomes larger. That is to say, we can control the rumor in a smaller range by choosing ℎ . From Fig. 9 and Fig. 10, we also obtain that the threshold value ℎ can change the stability of system (1). In addition, Fig. 8(b), Fig. 9(b), and Fig. 10(b) show that the system converges to the equilibrium in finite time. Indeed, this is a unique property of a discontinuous system that a smooth ODE system can not have, and using this feature, we can quickly and effectively control the spread of rumors.

Some comparison results
To demonstrate the significance of the discontinuous control strategy in system (1), we compare it to Ref. [18], which has a similar system to system (1) but with a saturated control function 1 We choose the following parameters = 0.4, = 0.5, Λ = 0.6, = 0.1, 1 = 0.4, 1 = 0.7, = 0, which are selected in Ref. [18]. Then, dynamics of system (1) without control strategies, with the saturated control and with the discontinuous control are shown in Fig. 11. From it, we obtain that by the effect of controller, the number of people spreading rumors has decreased, and the rumor spreaders are stable to a lower level under the discontinuous control. In addition, we also observed that when compared to saturation control, the system converges to equilibrium faster with discontinuous control. It demonstrates that by using a discontinuous controller, we can control the spread of rumors more quickly. Now we let > 0, from Ref. [18] we know that when above a threshold value system will be in a state of periodic oscillation, which makes it hard to stop the spread of the rumors. However, with a discontinuous control we can adjust the value of ℎ to control rumor quickly. Fig. 12 (a) shows that when = 2, system is in a periodic oscillation. With the saturation control system is still in the periodic oscillation (see Fig. 12(b)), which means that saturation control fails to effectively control the spread of rumors. However, Fig. 12(c) shows that rumors were quickly brought under control with discontinuous control.

Conclusions
In this study, we examine a model for the spread of rumors that involves a threshold that can be set arbitrarily high or low. We first performed a dynamic analysis of the two parts of the system and derived the necessary conditions for the local stability of the equilibrium states. Hopf bifurcation at the stable equilibrium was investigated with as the bifurcation parameter. Theoretical results and simulations showed that is what causes the model to switch between stable and unstable states, and Hopf bifurcation happens when goes above a threshold.
We also studied the sliding domain and its dynamics, including the existence of the tangent equilibrium, boundary equilibrium, regular equilibrium, and the stability of the pseudo-equilibrium.
Indeed, the situation of people in the social world may become more complicated when rumors break out, and this can influence the rumor's spread. We plan to further explore and improve the model of rumor spread and to provide more practical control techniques in the future. In addition, we mentioned how the complex network theory became more frequently used in the research of rumor spread thanks to Watts's WS small world network model [28] and Barabasi's BA scale-free model. It, we'll think about how to stop rumors from spreading on social media and analyze the techniques that people use to do so.

Author contribution statement
Chunru Li: Conceived and designed the experiments; Analyzed and interpreted the data; Wrote the paper. Zujun Ma: Conceived and designed the experiments; Contributed reagents, materials, analysis tools or data. Yuanyuan Wang: Performed the experiments; Analyzed and interpreted the data.

Funding statement
Zujun Man was supported by National Natural Science Foundation of China [71672154].

Data availability statement
Data will be made available on request.